Analytical impulse response of a fractional second order filter and its impulse response invariant discretization

doi:10.1016/j.sigpro.2010.01.017

Signal Processing

Volume 91, Issue 3, March 2011, Pages 498-507

https://doi.org/10.1016/j.sigpro.2010.01.017 Get rights and content

Abstract

In this paper, we derive the impulse response of a fractional second order filter of the form $(s^{2} + as + b)^{- γ}$ , where $a, b ⩾ 0$ and $γ > 0$ . The asymptotic properties of the impulse responses are obtained. Moreover, based on the derived analytical impulse response, we show how to perform the discretization of the above fractional second order filter. Finally, a number of illustrated examples in time and frequency domains are provided as proofs of concepts.

Introduction

Fractional calculus is a mathematical discipline which deals with derivatives and integrals of arbitrary real or complex orders [1], [2], [3], [4]. It was proposed more than 300 years ago and the theory was developed mainly in the 19th century. Several books [1], [2], [3], [4] provide a good source of references on fractional calculus. However, the applications of fractional calculus are just a recent focus of research. For pioneering works, we cite [5], [6], [7], [8].

The fractional order filter can be applied in signal modeling, filter design, controller design and nonlinear system identification [9], [10]. The key step toward the application of the fractional order filter is its numerical discretization. The conventional discretization method for fractional order filter is the frequency-domain fitting technique (indirect method). In indirect discretization methods [11], two steps are required, i.e., frequency domain fitting in continuous time domain first and then discretizing the fit s-transfer function. Other frequency-domain fitting methods can also be used but without guaranteeing the stable minimum-phase discretization [12]. In this paper, the direct discretization method will be used. An effective impulse response invariant discretization method was discussed in [12], [13], [14], [15]. The method is a technique for designing discrete-time infinite impulse response (IIR) filters from continuous-time fractional order filters in which the impulse response of the continuous-time fractional order filter is sampled to produce the impulse response of the discrete-time filter. For more discussions of discretization methods, we cite [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

The physical realization of $(s^{2} + as + b)^{- γ}$ can be illustrated as the type II fractional Langevin equation describing the fractional oscillator process with two indices [22]. The centered stationary formula discussed in [22] can be partly extended by using the discussions in this paper. For other previous works, we cite [19], [20], [21].

In this paper, we first focus on the inverse Laplace transform of $(s^{2} + as + b)^{- γ}$ by cutting the complex plane and computing the complex integrals. The derived results can be easily computed in Matlab and applied to obtain the asymptotic properties of the continuous impulse responses. Moreover, a direct discretization method is used to get the digital impulse responses. The results are compared in both of the time and frequency domains. Lastly, several figures are provided as proof of concepts.

The following part of this paper is organized as: In Section 2, the basic mathematical tools are introduced. In Section 3, the time domain analysis is derived, in which the continuous impulse response and its asymptotic properties are obtained. The time and frequency responses are shown in Section 4. The conclusions and future works are discussed in Section 5.

Section snippets

Laplace transform and Z transform

The Laplace transform of a function f(t), defined for all real numbers $t \geq 0$ , is the function F(s), defined by $F (s) = L {f (t)} = \int_{0}^{\infty} e^{- st} f (t) d t .$

The inverse Laplace transform is given by the following complex integral: $f (t) = L^{- 1} {F (s)} = \frac{1}{2 π i} \int_{σ - i \infty}^{σ + i \infty} e^{st} F (s) d s,$ where $σ$ is a real number so that the contour path of integration is in the region of convergence of F(s) normally requiring $σ > Re {s_{k}}$ for every singularity s_k of F(s) and i²=−1 [23].

The single-sided Z transform of a discrete-time signal x[n] is the

Derivation of the analytical impulse response of $(s^{2} + as + b)^{- γ}$

In this section, the inverse Laplace transform of $(s^{2} + as + b)^{- γ} = L {g (t)}$ is derived by using the complex integral which can lead to some useful asymptotic properties of g(t).

Let $G (s) = \frac{1}{(s^{2} + as + b)^{γ}},$ where $a, b \geq 0$ , $γ > 0$ and $L {g (t)} = G (s)$ . It can be seen that there are two poles of G(s), $s_{1} = (- a - \sqrt{a^{2} - 4 b}) / 2$ and $s_{2} = (- a + \sqrt{a^{2} - 4 b}) / 2$ . It follows that $G (s) = \frac{1}{(s - s_{1})^{γ}} \cdot \frac{1}{(s - s_{2})^{γ}} .$

Let $c \in {s_{1}, s_{2}}$ , $L^{- 1} \{\frac{1}{(s - c)^{γ}}\} = \frac{1}{2 π i} \int_{σ - i \infty}^{σ + i \infty} \frac{e^{st}}{(s - c)^{γ}} d s .$ When $γ \in {γ | γ > 0, γ \neq 1, 2, 3, \dots}$ , we have s=c and $s = \infty$ are the two branch points of $e^{st} (s - c)^{- γ}$ . It follows

Impulse response invariant discretization of $(s^{2} + as + b)^{- γ}$

Based on the obtained analytical impulse response function g(t), given sampling period T_s, it is straightforward to perform the inverse response invariant discretization of $(s^{2} + as + b)^{- γ}$ by using the Prony technique [18], [30], [31] which is an algorithm for finding an IIR filter with a prescribed time domain impulse response. It has applications in filter design, exponential signal modeling, and system identification (parametric modeling) [30], [31].

The plots of g(t) for different a, b and $γ$ are

Conclusions and future works

In this paper, we first discussed the continuous impulse responses of fractional order filter $G (s) = (s^{2} + as + b)^{- γ}$ and their asymptotic properties. It was shown that the characters of g(t) were strongly related to the poles of G(s), such as the oscillations happened only for $a^{2} - 4 b < 0$ and the decaying speed was determined by a and b. The Laplace final value and initial value theorems can be used to verify the limit properties of g(t) for $a, b > 0$ . Moreover, the impulse response invariant discretization

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Signal Processing

Analytical impulse response of a fractional second order filter and its impulse response invariant discretization

Abstract

Introduction

Section snippets

Laplace transform and Z transform

Derivation of the analytical impulse response of (s2+as+b)−γ

Impulse response invariant discretization of (s2+as+b)−γ

Conclusions and future works

The Fractional Calculus

An Introduction to the Fractional Calculus and Fractional Differential Equations

Fractional Differential Equations

Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204

A theoretical basis for the application of fractional calculus to viscoelasticity

Journal of Rheology

Fractional-order systems and PIλDμ-controllers

IEEE Transactions on Automatic Control

Fractional Schrödinger equation

Physical Review E

Analysis and design of fractional-order digital control systems

Systems Analysis Modelling Simulation

Frequency-band complex noninteger differentiator: characterization and synthesis

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications

Discretization schemes for fractional-order differentiators and integrators

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications

A new IIR-type digital fractional order differentiator

Signal Processing

Discretized fractional calculus

SIAM Journal on Mathematical Analysis

Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review

Nonlinear Dynamics

Two direct Tustin discretization methods for fractional-order differentiator/integrator

Journal of The Franklin Institute

Implementation of discrete-time fractional-order controllers based on LS approximations

Acta Polytechnica Hungarica

Impulse invariance-based method for the computation of fractional integral of order 0<α<1

Computers and Electrical Engineering

Derivation of the analytical impulse response of $(s^{2} + as + b)^{- γ}$

Impulse response invariant discretization of $(s^{2} + as + b)^{- γ}$

Fractional-order systems and ${PI}^{λ} D^{μ} -controllers$

Impulse invariance-based method for the computation of fractional integral of order $0 < α < 1$